Universal tools for analysing structures and interactions in geometry

This study examined symmetry and perspective in modern geometric transformations, treating them as functions that preserve specific properties while mapping one geometric figure to another. The purpose of this study was to investigate geometric transformations as a tool for analysis, to consider invariants as universal tools for studying geometry. Materials and Methods: The Erlangen ideas of F. I. Klein were used, which consider geometry as a theory of group invariants with respect to the transformation of the plane and space. Results and Discussion: Projective transformations and their extension to two-dimensional primitives were investigated. Two types of geometric correspondences, collinearity and correlation, and their properties were studied. The group of homotheties, including translations and parallel translations, and their role in the affine group were investigated. Homology with ideal line axes, such as stretching and centre stretching, was considered. Involutional homology and harmonic homology with the centre, axis, and homologous pairs of points were investigated. In this study unified geometry concepts, exploring how different geometric transformations relate and maintain properties across diverse geometric systems. Conclusions: It specifically examined Möbius transforms, including their matrix representation, trace, fixed points, and categorized them into identical transforms, nonlinear transforms, shifts, dilations, and inversions

Shape localization, quantification and correspondence using Region Matching Algorithm

We propose a method for local, region-based matching of planar shapes, especially as those shapes that change over time. This is a problem fundamental to medical imaging, specifically the comparison over time of mammograms. The method is based on the non-emergence and non-enhancement of maxima, as well as the causality principle of integral invariant scale space. The core idea of our Region Matching Algorithm (RMA) is to divide a shape into a number of “salient” regions and then to compare all such regions for local similarity in order to quantitatively identify new growths or partial/complete occlusions. The algorithm has several advantages over commonly used methods for shape comparison of segmented regions. First, it provides improved key-point alignment for optimal shape correspondence. Second, it identifies localized changes such as new growths as well as complete/partial occlusion in corresponding regions by dividing the segmented region into sub-regions based upon the extrema that persist over a sufficient range of scales. Third, the algorithm does not depend upon the spatial locations of mammographic features and eliminates the need for registration to identify salient changes over time. Finally, the algorithm is fast to compute and requires no human intervention. We apply the method to temporal pairs of mammograms in order to detect potentially important differences between them

Shape description and matching using integral invariants on eccentricity transformed images

Matching occluded and noisy shapes is a problem frequently encountered in medical image analysis and more generally in computer vision. To keep track of changes inside the breast, for example, it is important for a computer aided detection system to establish correspondences between regions of interest. Shape transformations, computed both with integral invariants (II) and with geodesic distance, yield signatures that are invariant to isometric deformations, such as bending and articulations. Integral invariants describe the boundaries of planar shapes. However, they provide no information about where a particular feature lies on the boundary with regard to the overall shape structure. Conversely, eccentricity transforms (Ecc) can match shapes by signatures of geodesic distance histograms based on information from inside the shape; but they ignore the boundary information. We describe a method that combines the boundary signature of a shape obtained from II and structural information from the Ecc to yield results that improve on them separately

Recognition of occluded objects using curvature

New approaches of object representation reliable for partially occluded objects recognition are introduced in this article. Objects are represented by their boundaries, which are deformed by the occlusion. The boundary representation was made by approximation with circle arcs. The representation was designed to be local and robust to occlusion. The curve approximation with circle arcs is equivalent to the curvature representation with respect to noise. The algorithm is simple and easy to implement. Experimental results are presented

Joint shape and motion estimation from echo-based sensor data

2018 Fall.Includes bibliographical references.Given a set of time-series data collected from echo-based ranging sensors, we study the problem of jointly estimating the shape and motion of the target under observation when the sensor positions are also unknown. Using an approach first described by Stuff et al., we model the target as a point configuration in Euclidean space and estimate geometric invariants of the configuration. The geometric invariants allow us to estimate the target shape, from which we can estimate the motion of the target relative to the sensor position. This work will unify the various geometric- invariant based shape and motion estimation literature under a common framework, and extend that framework to include results for passive, bistatic sensor systems

View-Invariance in Visual Human Motion Analysis

This thesis makes contributions towards the solutions to two problems in the area of visual human motion analysis: human action recognition and human body pose estimation. Although there has been a substantial amount of research addressing these two problems in the past, the important issue of viewpoint invariance in the representation and recognition of poses and actions has received relatively scarce attention, and forms a key goal of this thesis. Drawing on results from 2D projective invariance theory and 3D mutual invariants, we present three different approaches of varying degrees of generality, for human action representation and recognition. A detailed analysis of the approaches reveals key challenges, which are circumvented by enforcing spatial and temporal coherency constraints. An extensive performance evaluation of the approaches on 2D projections of motion capture data and manually segmented real image sequences demonstrates that in addition to viewpoint changes, the approaches are able to handle well, varying speeds of execution of actions (and hence different frame rates of the video), different subjects and minor variabilities in the spatiotemporal dynamics of the action. Next, we present a method for recovering the body-centric coordinates of key joints and parts of a canonically scaled human body, given an image of the body and the point correspondences of specific body joints in an image. This problem is difficult to solve because of body articulation and perspective effects. To make the problem tractable, previous researchers have resorted to restricting the camera model or requiring an unrealistic number of point correspondences, both of which are more restrictive than necessary. We present a solution for the general case of a perspective uncalibrated camera. Our method requires that the torso does not twist considerably, an assumption that is usually satisfied for many poses of the body. We evaluate the quantitative performance of the method on synthetic data and the qualitative performance of the method on real images taken with unknown cameras and viewpoints. Both these evaluations show the effectiveness of the method at recovering the pose of the human body