7,589 research outputs found
M\"obius Invariants of Shapes and Images
Identifying when different images are of the same object despite changes
caused by imaging technologies, or processes such as growth, has many
applications in fields such as computer vision and biological image analysis.
One approach to this problem is to identify the group of possible
transformations of the object and to find invariants to the action of that
group, meaning that the object has the same values of the invariants despite
the action of the group. In this paper we study the invariants of planar shapes
and images under the M\"obius group , which arises
in the conformal camera model of vision and may also correspond to neurological
aspects of vision, such as grouping of lines and circles. We survey properties
of invariants that are important in applications, and the known M\"obius
invariants, and then develop an algorithm by which shapes can be recognised
that is M\"obius- and reparametrization-invariant, numerically stable, and
robust to noise. We demonstrate the efficacy of this new invariant approach on
sets of curves, and then develop a M\"obius-invariant signature of grey-scale
images
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 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
Forward scatter radar for air surveillance: Characterizing the target-receiver transition from far-field to near-field regions
A generalized electromagnetic model is presented in order to predict the response of forward scatter radar (FSR) systems for air-target surveillance applications in both far-field and near-field conditions. The relevant scattering problem is tackled by developing the Helmholtz-Kirchhoff formula and Babinet's principle to express the scattered and the total fields in typical FSR configurations. To fix the distinctive features of this class of problems, our approach is applied here to metallic targets with canonical rectangular shapes illuminated by a plane wave, but the model can straightforwardly be used to account for more general scenarios. By exploiting suitable approximations, a simple analytical formulation is derived allowing us to efficiently describe the characteristics of the FSR response for a target transitioning with respect to the receiver from far-field to near-field regions. The effects of different target electrical sizes and detection distances on the received signal, as well as the impact of the trajectory of the moving object, are evaluated and discussed. All of the results are shown in terms of quantities normalized to the wavelength and can be generalized to different configurations once the carrier frequency of the FSR system is set. The range of validity of the proposed closed-form approach has been checked by means of numerical analyses, involving comparisons also with a customized implementation of a full-wave commercial CAD tool. The outcomes of this study can pave the way for significant extensions on the applicability of the FSR technique
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
Detecting Similarity of Rational Plane Curves
A novel and deterministic algorithm is presented to detect whether two given
rational plane curves are related by means of a similarity, which is a central
question in Pattern Recognition. As a by-product it finds all such
similarities, and the particular case of equal curves yields all symmetries. A
complete theoretical description of the method is provided, and the method has
been implemented and tested in the Sage system for curves of moderate degrees.Comment: 22 page
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